About 28%. There is about a 28% chance that a random sample of 100 adults will have a sample proportion with more than 0.05 error in either direction. In other words, there is about a 28% chance that sample proportions fall below 0.25 or above 0.35 if the true population proportion is 0.30.
Here is how we got this probability.
Check normality conditions:
Conditions are met. In a sample of 100, we expect 30% successes and 70% failures. 100(0.30) = 30 and 100(0.70) = 70. So we can use a normal model.
Find the Z-score:
We want the error to be more than 5%. So the sample proportion could be less than 0.25 or greater than 0.35. It does not matter which sample proportion we use to find the Z-score because of the symmetry in the distribution. We arbitrarily chose 0.35. You could also have used 0.25.
Because of the symmetry in the distribution, we know that the Z-score for a sample proportion of 0.25 is -1.09.
We want the probability described by the two tails. The probability for one tail is 0.1379, or about 0.14. So the probability for both tails is about 2 x 0.14 = 0.28.
Conclusion: If it is true that 30% of smokers started smoking before the age of 16, then there is a 28% chance that the percentage from a random sample of 100 smokers is off by more than 5%. In other words, there is a 28% chance that the sample proportion is either less than 0.25 or greater than 0.35.